How can I overlay snow onto a static scene to make an animation?
Some tips that might help: 


If you want visual depth, then some snow has to "dissapear" behind the
castle (So the snow density in front of the castle is much lower). 
If i understand your current setup correctly this doesn't happen.
If you put the snow and the castle on different layers be sure to
tick mask for the castle layer on the snow layer (https://docs.blender.org/manual/en/latest/render/post_process/layers.html)
Also make sure your particle emitter Geometry has enough depth.
(horizontal plane).
Then it might be nice to add some force fields (to add some vortices
etc. you might add keyframes to the strength of some force fields and
use noise
How to make flickering light?
...


PS: You can loop your particles like this: https://m.youtube.com/watch?v=yanQ1C7GB04